Integrand size = 21, antiderivative size = 169 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {d (2 b c-5 a d) (4 b c-3 a d) x \sqrt {a+b x^2}}{8 a b^3}-\frac {d (4 b c-5 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{a b \sqrt {a+b x^2}}+\frac {3 d \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{7/2}} \]
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Time = 0.14 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {424, 542, 396, 223, 212} \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {3 d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right )}{8 b^{7/2}}-\frac {d x \sqrt {a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{8 a b^3}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) (4 b c-5 a d)}{4 a b^2}+\frac {x \left (c+d x^2\right )^2 (b c-a d)}{a b \sqrt {a+b x^2}} \]
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Rule 212
Rule 223
Rule 396
Rule 424
Rule 542
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) x \left (c+d x^2\right )^2}{a b \sqrt {a+b x^2}}+\frac {\int \frac {\left (c+d x^2\right ) \left (a c d-d (4 b c-5 a d) x^2\right )}{\sqrt {a+b x^2}} \, dx}{a b} \\ & = -\frac {d (4 b c-5 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{a b \sqrt {a+b x^2}}+\frac {\int \frac {a c d (8 b c-5 a d)-d (2 b c-5 a d) (4 b c-3 a d) x^2}{\sqrt {a+b x^2}} \, dx}{4 a b^2} \\ & = -\frac {d (2 b c-5 a d) (4 b c-3 a d) x \sqrt {a+b x^2}}{8 a b^3}-\frac {d (4 b c-5 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{a b \sqrt {a+b x^2}}+\frac {\left (3 d \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^3} \\ & = -\frac {d (2 b c-5 a d) (4 b c-3 a d) x \sqrt {a+b x^2}}{8 a b^3}-\frac {d (4 b c-5 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{a b \sqrt {a+b x^2}}+\frac {\left (3 d \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^3} \\ & = -\frac {d (2 b c-5 a d) (4 b c-3 a d) x \sqrt {a+b x^2}}{8 a b^3}-\frac {d (4 b c-5 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{a b \sqrt {a+b x^2}}+\frac {3 d \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{7/2}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.82 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x \left (8 b^3 c^3-15 a^3 d^3+a^2 b d^2 \left (36 c-5 d x^2\right )+2 a b^2 d \left (-12 c^2+6 c d x^2+d^2 x^4\right )\right )}{8 a b^3 \sqrt {a+b x^2}}-\frac {3 d \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 b^{7/2}} \]
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Time = 2.43 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {\frac {15 \sqrt {b \,x^{2}+a}\, a d \left (a^{2} d^{2}-\frac {12}{5} a b c d +\frac {8}{5} b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{8}-\frac {15 x \left (\frac {8 d \left (-\frac {1}{12} d^{2} x^{4}-\frac {1}{2} c d \,x^{2}+c^{2}\right ) a \,b^{\frac {5}{2}}}{5}-\frac {12 \left (-\frac {5 d \,x^{2}}{36}+c \right ) d^{2} a^{2} b^{\frac {3}{2}}}{5}+\sqrt {b}\, a^{3} d^{3}-\frac {8 b^{\frac {7}{2}} c^{3}}{15}\right )}{8}}{a \,b^{\frac {7}{2}} \sqrt {b \,x^{2}+a}}\) | \(137\) |
risch | \(-\frac {x \,d^{2} \left (-2 b d \,x^{2}+7 a d -12 b c \right ) \sqrt {b \,x^{2}+a}}{8 b^{3}}+\frac {\frac {7 a^{2} d^{3} x}{\sqrt {b \,x^{2}+a}}+\frac {8 b^{3} c^{3} x}{a \sqrt {b \,x^{2}+a}}-\frac {12 a b c \,d^{2} x}{\sqrt {b \,x^{2}+a}}+\left (15 a^{2} b \,d^{3}-36 a \,b^{2} c \,d^{2}+24 b^{3} c^{2} d \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{8 b^{3}}\) | \(165\) |
default | \(\frac {c^{3} x}{a \sqrt {b \,x^{2}+a}}+d^{3} \left (\frac {x^{5}}{4 b \sqrt {b \,x^{2}+a}}-\frac {5 a \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )}{4 b}\right )+3 c \,d^{2} \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+3 c^{2} d \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )\) | \(215\) |
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Time = 0.29 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.46 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (8 \, a^{2} b^{2} c^{2} d - 12 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} + {\left (8 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, a b^{3} d^{3} x^{5} + {\left (12 \, a b^{3} c d^{2} - 5 \, a^{2} b^{2} d^{3}\right )} x^{3} + {\left (8 \, b^{4} c^{3} - 24 \, a b^{3} c^{2} d + 36 \, a^{2} b^{2} c d^{2} - 15 \, a^{3} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (a b^{5} x^{2} + a^{2} b^{4}\right )}}, -\frac {3 \, {\left (8 \, a^{2} b^{2} c^{2} d - 12 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} + {\left (8 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, a b^{3} d^{3} x^{5} + {\left (12 \, a b^{3} c d^{2} - 5 \, a^{2} b^{2} d^{3}\right )} x^{3} + {\left (8 \, b^{4} c^{3} - 24 \, a b^{3} c^{2} d + 36 \, a^{2} b^{2} c d^{2} - 15 \, a^{3} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{8 \, {\left (a b^{5} x^{2} + a^{2} b^{4}\right )}}\right ] \]
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\[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{3}}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.17 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {d^{3} x^{5}}{4 \, \sqrt {b x^{2} + a} b} + \frac {3 \, c d^{2} x^{3}}{2 \, \sqrt {b x^{2} + a} b} - \frac {5 \, a d^{3} x^{3}}{8 \, \sqrt {b x^{2} + a} b^{2}} + \frac {c^{3} x}{\sqrt {b x^{2} + a} a} - \frac {3 \, c^{2} d x}{\sqrt {b x^{2} + a} b} + \frac {9 \, a c d^{2} x}{2 \, \sqrt {b x^{2} + a} b^{2}} - \frac {15 \, a^{2} d^{3} x}{8 \, \sqrt {b x^{2} + a} b^{3}} + \frac {3 \, c^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} - \frac {9 \, a c d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} + \frac {15 \, a^{2} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {7}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {2 \, d^{3} x^{2}}{b} + \frac {12 \, a b^{4} c d^{2} - 5 \, a^{2} b^{3} d^{3}}{a b^{5}}\right )} x^{2} + \frac {8 \, b^{5} c^{3} - 24 \, a b^{4} c^{2} d + 36 \, a^{2} b^{3} c d^{2} - 15 \, a^{3} b^{2} d^{3}}{a b^{5}}\right )} x}{8 \, \sqrt {b x^{2} + a}} - \frac {3 \, {\left (8 \, b^{2} c^{2} d - 12 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^3}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
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